Is it true that the expansion of the universe means that stars in our Milky Way Galaxy are getting increasingly farther apart?

Similar to how planets are held in place around their respective stars and moons around planets, cosmic expansion only overcomes gravity on intergalactic distance scales. Within a galaxy, gravity holds the stars together against the cosmic expansion because they are simply too close together for cosmic expansion to "beat" gravity.

Galaxies can have relative motion other than the cosmic expansion; for example, Andromeda is moving towards the Milky Way on its way to a collision (in about 4 billion years). Not even all pairs of galaxies are moving away from one another.

The stars in a galaxy, like the Milky Way, are held together by gravitational forces within the galaxy; the expansion of the universe affects the large-scale structure of the universe between galaxies, resulting in increased distances on cosmological scales.